Problems like
[beautiful math coming... please be patient]$(2) + (3) = 5$
and
[beautiful math coming... please be patient]$\,(3) + 5 = 2\,$ are easy for some people and hard for others.
If they're easy for you, then jump right to the exercises.
Otherwise, read on—and keep in mind that explaining something simple, in words, often ends up sounding
very complicated!
SIGNED NUMBERS:
The phrase signed numbers refers to numbers that can be
either positive (like
[beautiful math coming... please be patient]$\,5\,$) or negative (like
[beautiful math coming... please be patient]$\,5\,$).
That is, signed numbers are allowed
to have a minus sign.
Every real number can be interpreted in two ways:
 as a position on a number line;
 as a movement.
Both of these interpretations—position and movement—are used when learning to add
and subtract signed numbers.
(Once you know how to do it, the process will become automatic and you probably won't think
about the position and movement stuff.)
NUMBERS AS POSITION:
The number
[beautiful math coming... please be patient]$\,3\,$ can mean:
go to position
[beautiful math coming... please be patient]$\,3\,$ on the number line.
The number
[beautiful math coming... please be patient]$\,3\,$ can mean:
go to position
[beautiful math coming... please be patient]$\,3\,$ on the number line.
NUMBERS AS MOVEMENT:
Positive numbers can indicate movement to the right.
For example,
[beautiful math coming... please be patient]$\,3\,$ can mean:
move
[beautiful math coming... please be patient]$\,3\,$ units to the right.
Negative numbers can indicate movement to the left.
For example,
[beautiful math coming... please be patient]$\,3\,$ can mean:
move
[beautiful math coming... please be patient]$\,3\,$ units to the left.
ADDING A NEGATIVE NUMBER:
When you add a negative number, you should put it in parentheses, unless it comes first.
For example, the sum of
[beautiful math coming... please be patient]$\,3\,$ and
[beautiful math coming... please be patient]$\,1\,$ should be written as
[beautiful math coming... please be patient]$\,3 + (1)\,$.
(Recall that the word sum refers to an addition problem.)
If you want, you can optionally put that first negative number in parentheses, too:
[beautiful math coming... please be patient]$\,(3) + (1)$.
SIZE VERSUS SIGN:
Every number has a size (its distance from zero).
Every nonzero number has a sign (positive or negative).
For example:
The number
[beautiful math coming... please be patient]$\,3\,$:
its size is
[beautiful math coming... please be patient]$\,3\,$, and its sign is positive.
The number
[beautiful math coming... please be patient]$\,3\,$:
its size is
[beautiful math coming... please be patient]$\,3\,$, and its sign is negative.
In the movement interpretation of a real number,
the size tells us how far to move, and
the sign tells us which direction to move.
USING POSITION/MOVEMENT IDEAS IN AN ADDITION PROBLEM:
Now we're ready to combine the position and
movement ideas in an addition problem.
The process is illustrated
first with an example:
Consider the problem:
[beautiful math coming... please be patient]$\,2 + (3) + 5\,$
 The first number,
[beautiful math coming... please be patient]$\,2\,$, indicates a position.
Go to
[beautiful math coming... please be patient]$\,2\,$ on the number line.
 Adding a negative number indicates movement to the left.
Thus, adding
[beautiful math coming... please be patient]$\,3\,$ says to move
[beautiful math coming... please be patient]$\,3\,$ units to the left.
 Adding a positive number indicates movement to the right.
Thus, adding
[beautiful math coming... please be patient]$\,5\,$ says to move
[beautiful math coming... please be patient]$\,5\,$ units to the right.
 You end up at position
[beautiful math coming... please be patient]$\,4\,$.
Thus,
[beautiful math coming... please be patient]$\,2 + (3) + 5 = 4\,$.
THE ‘START AT ZERO’ INTERPRETATION:
Or, you can always start at zero!
That is, write
[beautiful math coming... please be patient]$2 + (3) + 5$ as
[beautiful math coming... please be patient]$0 + 2 + (3) + 5$ .
The first number indicates position, and the remaining numbers indicate movement.
Start at
[beautiful math coming... please be patient]$\,0\,$, move
[beautiful math coming... please be patient]$\,2\,$ to the right,
[beautiful math coming... please be patient]$\,3\,$ to the left, and
[beautiful math coming... please be patient]$\,5\,$ to the right, ending up at
[beautiful math coming... please be patient]$\,4\,$.
You should understand both interpretations, but in practice you can use whichever is more natural to you.
The start at zero interpretation is used in the following discussion.
You probably don't want to be drawing number lines every time you need to do an
addition of signed numbers problem.
The good news is that every problem—no matter how many numbers are involved—boils down to either
a twonumber addition problem, or a twonumber subtraction problem, which can then be done efficiently in your head.
Keep reading!
ADDING NUMBERS WITH THE SAME SIGNS:
When you add numbers with the same signs (both positive or both negative),
then in your head you do an addition problem.
Here are two examples:

[beautiful math coming... please be patient]$2 + 3$
(Both numbers are positive.)
Start at zero. Move to the right
[beautiful math coming... please be patient]$\,2\,$, then to the right
[beautiful math coming... please be patient]$\,3\,$. End up at
[beautiful math coming... please be patient]$\,5\,$.
Thus,
[beautiful math coming... please be patient]$2 + 3 = 5\,$.

[beautiful math coming... please be patient]
$(2) + (3)$
(Both numbers are negative.)
Start at zero. Move to the left
[beautiful math coming... please be patient]$\,2\,$, then to the left
[beautiful math coming... please be patient]$\,3\,$.
The total distance moved is
[beautiful math coming... please be patient]$\,2 + 3 = 5\,$.
You moved to the left, so you end up at
[beautiful math coming... please be patient]$\,5\,$.
Thus,
[beautiful math coming... please be patient]$\, (2) + (3) = 5\,$.
Notice that in both of these problems, you do an
addition problem in your head, which gives the total distance moved.
If you always move to the right, then the final answer is positive.
If you always move to the left, then the final answer is negative.
ADDING NUMBERS WITH DIFFERENT SIGNS:
When you add numbers with different signs (one positive, one negative),
then in your head you do a subtraction problem.
Here are two examples:

[beautiful math coming... please be patient]$2 + (3)$
(first number is positive, second number is negative)
Start at zero. Move
[beautiful math coming... please be patient]$\,2\,$ to the right and
[beautiful math coming... please be patient]$\,3\,$ to the left.
You moved more to the left—how much more?
Answer:
[beautiful math coming... please be patient]$\,3  2 = 1$
So you end up at
[beautiful math coming... please be patient]$\,1\,$.
Thus,
[beautiful math coming... please be patient]$\,2 + (3) = 1\,$.

[beautiful math coming... please be patient]$3 + (2)$
(first number is positive, second number is negative)
Start at zero. Move
[beautiful math coming... please be patient]$\,3\,$ to the right and
[beautiful math coming... please be patient]$\,2\,$ to the left.
You moved more to the right—how much more?
Answer:
[beautiful math coming... please be patient]$\,3  2 = 1\,$.
So you end up at
[beautiful math coming... please be patient]$\,1\,$.
Thus,
[beautiful math coming... please be patient]$\,3 + (2) = 1\,$.
The mental process is this:
Once you recognize that you're adding numbers with different signs,
throw away (for the moment) all the signs,
take the bigger number, and subtract the smaller number.
This gives you the
net distance traveled.
If you moved farther to the right, then your answer is positive.
If you moved farther to the left, then your answer is negative.
Notice that when you add numbers with different signs, then in your head you do a subtraction problem.
FIVESTEP PROCESS FOR ADDING TWO SIGNED NUMBERS:
When you add two signed numbers, you can follow this five step process.
As you read through these steps, think of applying these questions to the problem
[beautiful math coming... please be patient]$\,2 + (3)\,$:
 Step 1: What numbers are being added? (Answer:
[beautiful math coming... please be patient]$\,2\,$ and
[beautiful math coming... please be patient]$\,3\,$)
 Step 2: Do these numbers have the same sign or different signs? (Answer: different signs)
 Step 3: In your head, will you be doing an addition or subtraction problem? (Answer: subtraction problem)
 Step 4: Do the appropriate addition or subtraction problem.
Answer: Throw away the signs, leaving you with
[beautiful math coming... please be patient]$\,2\,$ and
[beautiful math coming... please be patient]$\,3\,$.
Subtract the smaller from the larger:
[beautiful math coming... please be patient]$\,3  2 = 1\,$
 Step 5: Is your answer positive or negative?
Answer: The bigger number is negative, so the answer will be negative.
So,
[beautiful math coming... please be patient]$\,2 + (3) = 1\,$.
MORE THAN TWO NUMBERS BEING ADDED:
If there are more than two numbers being added,
just turn it into a twonumber problem in the first step,
by combining the positive and negative numbers separately, like this:
[beautiful math coming... please be patient]$(3) + 5 + (2) + 1 + 4 + (6)$
[beautiful math coming... please be patient]$ = (\,(3) + (2) + (6)\,) \,+\, (5 + 1 + 4)$ (regroup, reorder, to combine negative and positive separately)
[beautiful math coming... please be patient]$ = 11 + 10$
[beautiful math coming... please be patient]$= 1$
Master the ideas from this section
by practicing
both exercises at the bottom of this page.
When you're done practicing, move on to:
Subtraction of Signed Numbers
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.